Combinatorics of the double-dimer model

TL;DR

This paper proves a new recurrence relation for the partition function of tripartite double-dimer configurations on planar bipartite graphs, extending previous results and exploring applications in tiling, cluster algebras, and algebraic geometry.

Contribution

It establishes a generalized recurrence relation for the double-dimer partition function, broadening the scope of previous identities and enabling new applications.

Findings

Proved a recurrence relation related to the Desnanot-Jacobi identity.

Extended work of Kenyon and Wilson to more general bipartite graphs.

Discussed potential applications in tiling theory, cluster algebras, and algebraic geometry.

Abstract

We prove that the partition function for tripartite double-dimer configurations of a planar bipartite graph satisfies a recurrence related to the Desnanot-Jacobi identity from linear algebra. A similar identity for the dimer partition function was established nearly 20 years ago by Kuo and has applications to random tiling theory and the theory of cluster algebras. This work was motivated in part by the potential for applications in these areas. Additionally, we discuss an application to Donaldson-Thomas and Pandharipande-Thomas theory which will be the subject of a forthcoming paper. The proof of our recurrence requires generalizing work of Kenyon and Wilson; specifically, lifting their assumption that the nodes of the graph are black and odd or white and even.